EVEQ 2008

International Summer School on Evolution Equations, Prague, Czech Republic, 16.–20. 6. 2008

Lectures

Grzegorz Karch

Nonlinear evolution equations with anomalous diffusion

Nonlinear and nonlocal evolution equations of the form u_t= -L u + f(u,∇ u), where L is a pseudodifferential operator representing the infinitesimal generator of a Lévy stochastic process and f is a smooth nonlinearity of polynomial growth, have been derived, for example, as models for growing interfaces in the case when the continuous Brownian diffusion surface transport is augmented by a random hopping mechanism. The goal of the lectures (based on the publications listed below) is to present recent results on properties of solutions to this equation resulting from the interplay between the strengths of the “diffusive” linear and “hyperbolic” nonlinear terms, posed in the whole space Rⁿ, and supplemented with suitable initial conditions.

References

1. P. Biler, G. Karch, and W.A. Woyczyński, Multifractal and Levy conservation laws, C. R. Acad. Sci. Paris, 330, Serie I, (2000) 343--348,
2. P. Biler, G. Karch, and W.A. Woyczyński, Asymptotics for conservation laws involving Levy diffusion generators, Studia Math. 148 (2001), 171--192.
3. P. Biler, G. Karch, and W.A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Levy conservation laws, Ann. I.H. Poincaré - Analyse non linéare 18, (2001), 613-637.
4. G. Karch and W.A. Woyczyński, Fractal Hamilton-Jacobi-KPZ equations, to appear in Trans. A.M.S.
5. G. Karch, Changxing Miao, and Xiaojing Xu, On convergence of solutions of fractal Burgers equation toward rarefaction waves, to appear in SIAM J. Math. Anal.

All papers are available on the webpage: http://www.math.uni.wroc.pl/~karch

Abstract in PDF can be found here (karch-abstract.pdf, ).

Presentation

• Nonlinear evolution equations with anomalous diffusion, Part I. Lévy operator (karch_notes_part_1.pdf, )
• Nonlinear evolution equations with anomalous diffusion, Part II. Fractal Burgers equations (karch_notes_part_2.pdf, )
• Nonlinear evolution equations with anomalous diffusion, Part III. Fractal Hamilton-Jacobi-KPZ equations (karch_notes_part_3.pdf, )

Josef Málek

Mathematical analysis of thermodynamics of incompressible fluids

The course has the following goals:

1. to outline the mathematical framework for unsteady flows of various classes of incompressible, heat conducting fluids (including both linear and nonlinear, explicit and implicit, continuous and discontinuous constitutive equations used to identify rheological properties of fluids),
2. to formulate problems concerning mathematical consistency of the models and to survey the available results concerning long-time and large-data properties of the solution (its existence, in particular),
3. to focus on identifying the mathematical methods involved in the analysis of the established results.

Presentation

• Mathematical analysis in thermodynamics of incompressible fluids (EVEQ2008_1.pdf, )
• On solvability of a non-linear heat equation with non-integrable convective term and the right-hand side involving measures (EVEQ2008_PartIII.pdf, )
• Results (EVEQ2008_Results.pdf, )

Laure Saint-Raymond

Hydrodynamic limits of the Boltzmann equation

The goal of that series of lectures is to present some mathematical results describing the transition from kinetic theory, and more precisely from the Boltzmann equation for perfect gases, to hydrodynamics. Different fluid asymptotics will be investigated starting from solutions of the Boltzmann equation which are only assumed to satisfy the estimates coming from physics, namely some bounds on mass, energy and entropy.

In that framework, the key mathematical tools for the study of hydrodynamic limits are

• entropy inequality which provides some uniform bounds on the size of the distribution (relative entropy), on its distance to local thermodynamic equi librium (entropy dissipation) and on the boundary data (Darrozès-Guiraud information);
• some decomposition of Chapman-Enskog's type which, combined with the smoothing effect of the collision operator, gives some additional integrability with res pect to velocity variables;
• some mixing properties of the free-transport operator (dispersion and averaging results) which enables - under suitable scaling assumptions - to gain strong compactness on the macroscopic fields.

We will consider incompressible regimes, both viscous and inviscid. In the first case, we implement all the previous tools and establish some optimal convergence resu lt towards the Navier-Stokes equations by a weak compactness method. In the second scaling, the last argument fails and we have to use rather a stability argument around smooth solutions of the incompressible Euler equations, the so-called relative entropy method - which requires in counterpart additional assumptions on the initial data.

Presentation

• The Boltzmann equations and its formal hydrodynamic limits (LSR1.pdf, )
• Mathematical tools for the study of hydrodynamic limits (LSR2.pdf, )
• Convergence results: from the Boltzmann equation to incompressible hydrodynamic models (LSR3.pdf, )

Giuseppe Savaré

A variational approach to gradient flows and rate-independent problems

We present a concise overview of the variational theory for gradient flows and for the "energetic formulation" of rate-independent processes. Starting from the classical formulation of these problems in linear (Hilbert/Banach) spaces, we will highlight their common metric structure and we will exploit this feature to derive some general existence, approximation, and stability results. Applications to various kind of nonlinear evolution PDE's will be given.

Presentation

• Gradient flows - metric formulation (Savare.pdf, )

Paolo Secchi

Free boundary problems for the equations of Fluid Dynamics of hyperbolic type

In this series of lectures we will consider some free boundary problems for the equations of motion of inviscid compressible flows in Fluid Dynamics and ideal MHD. In particular, we will focus on problems where the free boundary is a characteristic hypersurface and the Lopatinski condition for the associated linearized equations holds only in weak form.

The theory for this kind of problems is far from being complete. We will describe the state of the art and discuss some results and methods of approach.

Presentation

• Characteristic IBVP's of symmetric hyperbolic systems (EVEQPragaSenza.pdf, )
• 2D compressible vortex sheets (VortexSheetsSenza.pdf, )
• An example of loss of regularity (example.pdf, )

Denis Serre

Discrete shock profiles for systems of conservation laws

Hyperbolic systems of conservation laws cover a lot of phenomena of the real world, like gas dynamics, MHD, traffic flows and so on. A major feature is that their solutions develop discontinuities in finite time. These are called shock waves.

When approximating such a system by finite differences, one wishes that the scheme be stable and consistent. Consistency is the property that small disturbances of simple solutions are well-approximated. This includes the requirement that shock waves, which are traveling waves of the PDEs, be associated to traveling waves of the numerical scheme. The corresponding profile is called a Discrete Shock Profile (DSP). Despite the terminology, its domain is usually the whole real line, because the shock velocity s is not rationally dependent with the grid velocity λ:=Δ x/Δ t.

We shall recall basic definitions of conservation laws and of their discretization by conservative difference schemes. Then we shall discuss the existence of DSPs, and compare the situation with the approximation by viscosity (viscous shock profiles). We shall see that some obstruction may occur when η:=s/λ is irrational, except in the scalar case. We give proof of existence of a DSP when the shock is of Lax type with small strength, and the scheme has some dissipation.

Presentation

• Shock profiles in the numerical analysis of hyperbolic systems of conservation laws (Serre.pdf, )